Finite Group Databasebeta

← #481×D197 #38752608

Subgroups — #481×D197 #38752608

Order 25216.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×D197C211yesmaximal, characteristic
2C2×C4⋊C4×D197C211yesmaximal, characteristic, complemented
3#27161399C212yesmaximal, complemented
4#27161398C211yesmaximal, characteristic, complemented
5#27161397C211yesmaximal, characteristic, complemented
6#27161396C212yesmaximal, complemented
7#27161395C211yesmaximal, characteristic, complemented
8#106×D197C212yesmaximal, complemented
9#102×D197C212yesmaximal, complemented
10C22⋊C8×D197C211yesmaximal, characteristic, complemented
11#481×C197C211yesmaximal, characteristic, complemented
12C2×D4×D197—21no—
13C2×D4×D197C2211yescharacteristic
14C22×C4×D197—21no—
15C22×C4×D197C2211yescharacteristic
16C2×C8×D197C2212yes—
17#20263301C2211yescharacteristic
18#20263300C2211yescharacteristic
19#20263299—21no—
20#20263298—21no—
21#20263297C2211yescharacteristic
22C2×C4⋊Dic197C2211yescharacteristic, complemented
23C2×#15691337—21no—
24C4⋊C4×D197—21no—
25C4⋊C4×D197—21no—
26C4⋊C4×D197C2211yescharacteristic
27C4⋊C4×D197C2212yes—
28C4⋊C4×D197C2212yescomplemented
29C22⋊C4×D197—21no—
30#20263258C2212yescomplemented
31#20263257C2212yescomplemented
32#20263257C2212yescomplemented
33#20263248C2212yescomplemented
34#20263247C2212yescomplemented
35#20263246C2212yescomplemented
36C22⋊#13096802C2211yescharacteristic, complemented
37#20263238C2212yescomplemented
38#121×C197C2211yescharacteristic
39C2×C4⋊C4×C197C2211yescharacteristic, complemented
40#106×C197C2212yescomplemented
41#102×C197C2212yescomplemented
42C22⋊C8×C197C2211yescharacteristic, complemented
43C23×D197—21no—
44C23×D197D411yescharacteristic
45C2×D788—21no—
46C2×D788C2311yescharacteristic
47C2×C22⋊D197—21no—
48C2×C22⋊D197—21no—
49C2×C22⋊D197—21no—
50D4×D197—24no—
51D4×D197—44no—
52C22×Dic197—21no—
53C22×Dic197D411yescharacteristic
54C2×C4×D197—24no—
55C2×C4×D197—21no—
56C2×C4×D197—22no—
57C2×C4×D197—22no—
58C2×C4×D197—44no—
59C2×C4×D197D411yescharacteristic
60C2×C4×D197D411yescharacteristic
61C2×C4×D197C2311yescharacteristic
62C2×C4×D197C2311yescharacteristic
63C2×#13096802C2312yes—
64C8×D197—24no—
65C4⋊Dic197—21no—
66C4⋊Dic197C2312yes—
67C4⋊Dic197C2311yescharacteristic
68#15691337—22no—
69#15691337—22no—
70#15691337—21no—
71C22⋊Dic197—21no—
72#15691335—21no—
73#15691335—21no—
74C2×D4×C197—21no—
75C2×D4×C197C2311yescharacteristic
76C22×C4×C197—21no—
77C22×C4×C197C2311yescharacteristic
78C2×C8×C197C2312yes—
79C4⋊C4×C197—21no—
80C4⋊C4×C197C2312yes—
81C4⋊C4×C197C2311yescharacteristic
82C22⋊C4×C197—21no—
83C22×D197—22no—
84C22×D197—22no—
85C22×D197—21no—
86C22×D197—42no—
87C22×D197—42no—
88C22×D197—42no—
89C22×D197SD1614yes—
90C22×D197C2×D411yescharacteristic
91C22×D197C2×D411yescharacteristic
92D788—22no—
93D788—42no—
94C22⋊D197—42no—
95C22⋊D197—42no—
96C22⋊D197—42no—
97C2×Dic197—22no—
98C2×Dic197—21no—
99C2×Dic197—22no—
100C2×Dic197—42no—
101C2×Dic197C2×D411yescharacteristic
102C2×Dic197C2×D411yescharacteristic
103C4×D197—22no—
104C4×D197—22no—
105C4×D197—22no—
106C4×D197—44no—
107C4×D197—42no—
108C4×D197—44no—
109C4×D197#4114yes—
110#13096802—22no—
111C23×C197—21no—
112C23×C197C2×D411yescharacteristic
113D4×C197—22no—
114D4×C197—42no—
115C2×C4×C197—22no—
116C2×C4×C197—21no—
117C2×C4×C197—22no—
118C2×C4×C197—42no—
119C2×C4×C197C2×D411yescharacteristic
120C2×C4×C197C2411yescharacteristic
121C8×C197—22no—
122C22×C197—21no—
123C22×C197—21no—
124C22×C197—41no—
125C22×C197—41no—
126C22×C197—41no—
127C22×C197C2×SD1612yes—
128C22×C197C22×D411yescharacteristic
129C2×D197—21no—
130C2×D197—21no—
131C2×D197—28no—
132C2×D197—41no—
133C2×D197—41no—
134C2×D197—41no—
135C2×D197—84no—
136C2×D197#12312yes—
137C2×D197C2×SD1612yes—
138C2×D197C2×SD1612yes—
139C2×D197AGL(1,ℤ/8)12yes—
140C4×C197—21no—
141C4×C197—41no—
142C4×C197—42no—
143C4×C197C2×#4112yes—
144Dic197—21no—
145Dic197—41no—
146Dic197—42no—
147Dic197#12312yes—
148C2×C197—22no—
149C2×C197—81no—
150C2×C197C2×#12311yescharacteristic
151C2×C197C22×SD1611yescharacteristic
152C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
153D197—22no—
154D197—81no—
155D197#48114yesdirect product factor
156C197C2×#48111yescharacteristic, complemented, Sylow
157C2×#481—1971nomaximal, Sylow
158C2×#121—1971no—
159C22×C4⋊C4—1971no—
160#481—1971no—
161#481—1971no—
162#481—1972no—
163#481—1972no—
164#481—1971no—
165#481D19711yescharacteristic, direct product factor
166C2×#106—1972no—
167C2×#102—1972no—
168C2×C22⋊C8—1971no—
169C22×D4—3941no—
170C22×D4—1971no—
171C23×C4—3941no—
172C23×C4—1971no—
173C22×C8—1972no—
174#121—3941no—
175#121—3941no—
176#121—1971no—
177#121—1971no—
178#121—1971no—
179#121C2×D19711yescharacteristic
180C2×C4⋊C4—3941no—
181C2×C4⋊C4—3941no—
182C2×C4⋊C4—3941no—
183C2×C4⋊C4—1971no—
184C2×C4⋊C4—1972no—
185C2×C4⋊C4—1972no—
186C2×C4⋊C4—1971no—
187C2×C4⋊C4C2×D19711yescharacteristic, complemented
188C2×C22⋊C4—3941no—
189#106—1972no—
190#106—1972no—
191#106—1972no—
192#106C2×D19712yescomplemented
193#102—1972no—
194#102—1972no—
195#102—1972no—
196#102C2×D19712yescomplemented
197C22⋊C8—1971no—
198C22⋊C8—1972no—
199C22⋊C8C2×D19711yescharacteristic, complemented
200C24—3941no—
201C24—1971no—
202C2×D4—21no—
203C2×D4—7884no—
204C2×D4—3941no—
205C2×D4—3941no—
206C2×D4—3941no—
207C2×D4—3944no—
208C2×D4—3941no—
209C2×D4—1971no—
210C2×D4C22×D19711yescharacteristic
211C22×C4—21no—
212C22×C4—7884no—
213C22×C4—3941no—
214C22×C4—3942no—
215C22×C4—3942no—
216C22×C4—3944no—
217C22×C4—3941no—
218C22×C4—1971no—
219C22×C4—1971no—
220C22×C4—1971no—
221C22×C4—1971no—
222C22×C4—1971no—
223C22×C4C22×D19711yescharacteristic
224C2×C8—3944no—
225C2×C8—1972no—
226C2×C8C22×D19712yes—
227C4⋊C4—21no—
228C4⋊C4—3942no—
229C4⋊C4—3942no—
230C4⋊C4—3941no—
231C4⋊C4—3941no—
232C4⋊C4—1972no—
233C4⋊C4—1971no—
234C4⋊C4C22×D19712yes—
235C4⋊C4C22×D19711yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—3941no—
238C22⋊C4—3941no—
239C22⋊C4—3941no—
240C23—21no—
241C23—7882no—
242C23—7882no—
243C23—7882no—
244C23—3942no—
245C23—3942no—
246C23—3941no—
247C23—1974no—
248C23—1971no—
249C23—1971no—
250C23D4×D19711yescharacteristic
251D4—22no—
252D4—42no—
253D4—7882no—
254D4—7882no—
255D4—7882no—
256D4—7882no—
257D4—3942no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—7884no—
263C2×C4—7884no—
264C2×C4—7882no—
265C2×C4—7882no—
266C2×C4—3942no—
267C2×C4—3942no—
268C2×C4—3941no—
269C2×C4—3942no—
270C2×C4—3942no—
271C2×C4—3942no—
272C2×C4—1971no—
273C2×C4—1971no—
274C2×C4—1974no—
275C2×C4D4×D19711yescharacteristic
276C2×C4C23×D19711yescharacteristic
277C8—22no—
278C8—3942no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—7881no—
285C22—7881no—
286C22—7881no—
287C22—15764no—
288C22—3948no—
289C22—3941no—
290C22—3941no—
291C22—1972no—
292C22—1972no—
293C22—1972no—
294C22—1972no—
295C22SD16×D19712yes—
296C22C2×D4×D19711yescharacteristic
297C4—21no—
298C4—42no—
299C4—41no—
300C4—7882no—
301C4—7881no—
302C4—3941no—
303C4—1972no—
304C4#41×D19712yes—
305C2—22no—
306C2—81no—
307C2—15761no—
308C2—3942no—
309C2—1974no—
310C2#123×D19711yescharacteristic
311C2C2×SD16×D19711yescharacteristic
312C2AGL(1,ℤ/8)×D19711yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#481×C197C2
socleC22×C197C22×D4
prime radical#121×D197C2
hypercenter#481D197
hypocenterC197C2×#481
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4⋊C4×D197C2semidirect
3. #27161399C2semidirect
4. #27161398C2semidirect
5. #27161397C2semidirect
6. #27161396C2semidirect
7. #27161395C2semidirect
8. #106×D197C2semidirect
9. #102×D197C2semidirect
10. C22⋊C8×D197C2semidirect
11. #481×C197C2semidirect
22. C2×C4⋊Dic197C22semidirect
28. C4⋊C4×D197C22semidirect
30. #20263258C22semidirect
31. #20263257C22semidirect
32. #20263257C22semidirect
33. #20263248C22semidirect
34. #20263247C22semidirect
35. #20263246C22semidirect
36. C22⋊#13096802C22semidirect
37. #20263238C22semidirect
39. C2×C4⋊C4×C197C22semidirect
40. #106×C197C22semidirect
41. #102×C197C22semidirect
42. C22⋊C8×C197C22semidirect
155. D197#481direct (N × H)
156. C197C2×#481semidirect
165. #481D197direct (N × H)
187. C2×C4⋊C4C2×D197semidirect
192. #106C2×D197semidirect
196. #102C2×D197semidirect
199. C22⋊C8C2×D197semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#121×D1972
C2×C4⋊C4×D1972
#271613992
#271613982
#271613972
#271613962
#271613952
#106×D1972
#102×D1972
C22⋊C8×D1972
#481×C1972
C2×#481197